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... The origins of this book lie in our earlier book Random Processes: A Mathematical Approach for Engineers (Prentice Hall, 1986). This book began as
a second edition to the earlier book and the basic goal remains unchanged
– to introduce the fundamental ideas and mechanics of random processes to
engin ...

... Suppose G ∼ DP(α, H). G is a (random) probability measure over X.
We can treat it as a distribution over X. Let
θ1 , . . . , θ n ∼ G
be a random variable with distribution G.
We saw in the demo that draws from a Dirichlet
...

... the change from the current to the next point to be
approximately the same as the change from the immediately previous point to the current point and in the same
direction. Stated in a formula, u( j + 2) - u( j + 1) would
approximately equal u( j + 1) - u(j) with exact equality
for linear systems (w ...

... complexity. While they may integrate probability and logic successfully, it may
be very difficult to determine an answer to a question such as that of Schema (1).
Sometimes this is because a probabilistic logic seeks more generality than is required for applications; but often it is no fault of the ...

... Informally, σ(α) describes the rule for choosing a transition after α has occurred. The rule itself may be randomized. Since σ(α) is a sub-probability measure, it is possible that with some non-zero probability no transition is chosen,
which corresponds to terminating the computation (what in the pu ...

... information might become a curse rather than a blessing, as the more
information is available the more time is required to process it. In time critical
situations, time is an expensive commodity not always affordable. For
instance, consider a surgeon performing cardiac surgery. With all the new
adva ...

... The paper sets off to answer this question. In particular, in this paper we use
ideas from Computational Learning Theory to produce a strong set of guarantees
for this problem:
Objectives: We explicitly set our objective to be to distinguish attacking pairs
from non-attacking pairs, given our fairly ...

... representation language in its own right. It can be motivated in a number of
di
erent ways:
Determining what is in a system from observations (diagnosis and recognition) is an important part of AI. There have been many logic-based proposals
as to what is a diagnosis 17, 57, 13, 45, 12]. One problem ...

... n this module, we investigate the reliability of a component that can be repaired. In contrast to nonrepairable components, modeling of repairable components requires use of a random process. We begin
our effort in Section 12.2 by introducing renewal theory for the ordinary renewal process, as it is ...

... probability factors much less intuitive. Thus, despite their expressive power,
chain graph models have gained very little popularity as practical models for
decision support systems, and we shall therefore focus exclusively on models
that factorize according to DAGs.
As indicated above, probabilisti ...

... are studied, mainly focussing on mean value results as in [13]. Then, in Chapter 18, some
selected results of a single server queue with a general arrival process and general service times
are provided. Next, in Chapter 19, we extend our discussion to queueing networks. Finally,
in Chapter 20, stoch ...

... are studied, mainly focussing on mean value results as in [13]. Then, in Chapter 18, some
selected results of a single server queue with a general arrival process and general service times
are provided. Next, in Chapter 19, we extend our discussion to queueing networks. Finally,
in Chapter 20, stoch ...

... probabilities, p0 , p1 , . . . are the numbers satisfying (7.7) where {⌫i ; i 0} are the holdinginterval rates.
As one might guess, the appropriate approach to answering the above questions comes from
applying renewal theory to various renewal processes associated with the Markov process.
Many of th ...

... Euclidean distance is smaller than some threshold which, in general, is a function of N . In the
last two decades, this kind of random graphs was studied extensively – see the monograph of
Penrose [15] and the references therein. Numerous typical properties of such random graphs
have been investigat ...

... even if we restrict ourselves to stopping rules where the decision to
stop or continue is independent of future events, fairness may not be
preserved. For example, assume that Peter is allowed to go into debt
and can play as long as he wants to. He starts with 0 pennies and
decides to play until his ...

... is sufficiently reduced. For instance, Gao and Koller (2011) stop information gathering when
1) the conditional entropy of the interest variable is reduced beyond some threshold or 2)
the margin between the first and second most likely states of the interest variable is above
some threshold. In any ...

... Without the stochastic constraints on the vector entries, x is called a z eigenvector [Qi,
2005] or an l2 eigenvector [Lim, 2005] of P . Li and Ng [2014] and Gleich et al. [2015]
analyze when a solution vector x for Equation 1.4 exists and provide algorithms for
computing the vector. These algorithm ...

... construct such a system was purely logic based, and did not use probability. It gave the
correct interpretation and advice regarding the lumen in only 39% of cases. However,
when an appropriate Bayesian network was developed for the problem, the percentage of
correct results rose to over 90%. This i ...

... In subsequent sections we establish these insights in greater generality, a generality rich enough
to include many existing structural Markovian models of asset pricing. The framework for this
analysis, which allows for continuous state spaces and a richer information structure, is introduced
in Sec ...

... reporting of their information by each individual is sufficient for aggregation of information by a law of large numbers argument (Condorcet, 1788). Against this background,
a number of papers, most notably Bikchandani, Hirshleifer and Welch (1992), Banerjee
(1992) and Smith and Sorensen (2000), sho ...

... To answer the question of Example 1.1 we need to know the distribution of the random
variable X that denotes the number of Malus particles in a 2 liter sample from the Lake
Diarrhea. To fix the distribution of X we have to assume something about the distribution
of the Malus particles in the lake. W ...

... Asynchronous Transfer Mode (ATM), high-speed, cell-relay networks will most likely be
first used as backbones for the interconnection of enterprise networks composed of several LANs,
and may also carry VBR video traffic. A lot of studies have been made for the design, control and
performance of such ...

... The beeping model of network communication [1–3,10,14,19] assumes a collection of computational nodes,
connected in a network, that interact by beeping in synchronous rounds. If a node decides to beep in a given
round, it receives no feedback from the channel. On the other hand, if a node decides to ...

Stochastic geometry models of wireless networks

In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.In the early 1960s a pioneering stochastic geometry model was developed to study wireless networks. This model is considered to be the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireless network technologies including mobile ad hoc networks, sensor networks, vehicular ad hoc networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks. Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.The principal idea underlying the research of these stochastic geometry models, also known as random spatial models, is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.